Deterministic and Stochastic Models for Coalescence (aggregation, Coagulation): a Review of the Mean-field Theory for Probabilists

Consider N particles, which merge into clusters according to the rule: a cluster of size x and a cluster of size y merge at (stochastic) rate K (x; y)=N , where K is a speciied rate kernel. This Marcus-Lushnikov model of stochastic coalescence, and the underlying deter-ministic approximation given by the Smoluchowski coagulation equations , have an extensive scientiic literature. Some mathematical literature (Kingman's coalescent in population genetics; component sizes in random graphs) implicitly studies the special cases K (x; y) = 1 and K (x; y) = xy. We attempt a wide-ranging survey. General kernels are only now starting to be studied rigorously, so many interesting open problems appear.